On a Zero-Sum Generalization of a Variation of Schur’s Equation

摘要

Let $mgeqslant 3$ be a positive integer, and let $mathbb{Z}_m$ denote the cyclic group of residues modulo m. Furthermore, let $R(L_m; 2)(R(L_m;mathbb{Z}_m))$ denote the minimum integer N such that for every function $Delta: {1,2,ldots,N}rightarrow {0,1},(Delta: {1,2,ldots,N}rightarrow mathbb{Z}_m)$ there exist m integers $x_1< x_2